8slides(1) - Chapter 8: Hilbert Systems, Deduction Theorem...

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Unformatted text preview: Chapter 8: Hilbert Systems, Deduction Theorem Introduction Hilbert Systems The Hilbert proof systems are based on a language with implication and contain a Modus Ponens rule as a rule of inference. Modus Ponens is the oldest of all known rules of inference as it was already known to the Stoics (3rd century B.C.). It is also considered as the most natural to our intuitive thinking and the proof sys- tems containing it as the inference rule play a special role in logic. 1 Hilbert System H 1 : H 1 = ( L {} , F { A 1 ,A 2 } MP ) A1 ( A ( B A )) , A2 (( A ( B C )) (( A B ) ( A C ))) , MP ( MP ) A ; ( A B ) B , 2 Finding formal proofs in this system requires some ingenuity. The formal proof of ( A A ) in H 1 is a sequence B 1 , B 2 , B 2 , B 2 ,B 5 as defined below. B 1 = (( A (( A A ) A )) (( A ( A A )) ( A A ))) , axiom A2 for A = A , B = ( A A ), and C = A B 2 = ( A (( A A ) A )) , axiom A1 for A = A , B = ( A A ) B 3 = (( A ( A A )) ( A A ))), MP application to B 1 and B 2 B 4 = ( A ( A A )) , axiom A1 for A = A , B = A B 5 = ( A A ) MP application to B 3 and B 4 3 A general procedure for searching for proofs...
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8slides(1) - Chapter 8: Hilbert Systems, Deduction Theorem...

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